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A114336 Pythagorean triples of nearly isosceles triangle.

%I A114336 #56 Jul 13 2025 17:36:18
%S A114336 3,4,5,20,21,29,119,120,169,696,697,985,4059,4060,5741,23660,23661,
%T A114336 33461,137903,137904,195025,803760,803761,1136689,4684659,4684660,
%U A114336 6625109,27304196,27304197,38613965,159140519,159140520,225058681,927538920,927538921,1311738121,5406093003,5406093004,7645370045
%N A114336 Pythagorean triples of nearly isosceles triangle.
%C A114336 Pythagorean triples of exact isosceles triangles do not exist because 2a^2 = c^2 has no integer solution. a^2 + (a+1)^2 = c^2 are nearly isosceles triangles and give a recursive series.
%D A114336 Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
%H A114336 C.C. Chen and T.A. Peng, <a href="https://ajc.maths.uq.edu.au/pdf/11/ocr-ajc-v11-p263.pdf">Classroom note: Almost-isosceles right-angled triangles</a>, Australasian Journal of Combinatorics, Volume 11(1995), pp. 263-267. See p. 266.
%F A114336 a^2 + (a+1)^2 = c^2, a(n) = 3a(n-1) + 2c(n-1) + 1, c(n) = 4a(n-1) + 3c(n-1) + 2.
%F A114336 a(n) = (A002315(n) - 1)/2. - _Miguel-Ángel Pérez García-Ortega_, Nov 06 2024
%e A114336 119^2 + 120^2 = 169^2.
%e A114336 Triples begin:
%e A114336   n=1:   3,   4,   5;
%e A114336   n=2:  20,  21,  29;
%e A114336   n=3: 119, 120, 169;
%e A114336   n=4: 696, 697, 985;
%e A114336   ...
%t A114336 a=Table[(LucasL[2*n+1,2]-2)/4,{n,1,13}];Apply[Join,Map[{#,#+1,Sqrt[2#^2+2#+1]}&,a]] (* _Miguel-Ángel Pérez García-Ortega_, Nov 06 2024 *)
%o A114336 (BASIC)
%o A114336 a(1):= 3
%o A114336 c(1):= 5
%o A114336 for n:=2 until 10 step 1
%o A114336 a(n):= 3*a(n-1) + 2*c(n-1) + 1
%o A114336 c(n):= 4*a(n-1) + 3*c(n-1) + 2
%o A114336 print a(n),a(n)+1,c(n)
%o A114336 next n
%o A114336 end
%Y A114336 Cf. A001652, A001653, A046090.
%K A114336 easy,nonn,tabf
%O A114336 1,1
%A A114336 Heinrich Baldauf (heinbald25(AT)web.de), Feb 07 2006
%E A114336 More terms from _Robert Hutchins_, Jun 10 2009
%E A114336 More terms from _Miguel-Ángel Pérez García-Ortega_, Nov 06 2024